The most rapid possible growth of the maximum modulus of a canonical product of noninteger order with a prescribed majorant of the counting function of zeros (Q2848646)

This paper concerns the part of the theory of entire functions related to connections between the distribution of zeros of a function \(f\) and the growth rate of its maximum modulus \(M(f,r)=\max_{|z|\leqslant r}|f(z)|\). The author obtains asymptotic estimates for \(\log M(f,r)\) in the case when \(f\) is a canonical product and has noninteger order \(\rho>0\), and appropriate constraints hold for the number of zeros of \(f\) in the discs \(|z|\leqslant r\). These estimates are obtained in the case when the counting function of zeros has a prescribed majorant \(r^{\rho(r)}\), where \(\rho(r)\) is a proximate order such that one is continuous on \((R_0,+\infty)\), is in the class \(W^1_\infty\) on each compact interval of this ray and \(\lim_{r\to+\infty}\rho'(r)r\log r=~0\) exists over a subset of \((R_0,+\infty)\) whose complement has measure zero. In this paper the author extends the known Valiron-Gold'berg theorem along the lines tried by Valiron, although without success.NEWLINENEWLINELet \(\rho(r)\), \(\lim_{r\to\infty}\rho(r)=\rho>0\), be an arbitrary proximate order and \(\Lambda=\{\lambda_n\}_{n\in\mathbb N}\) (\(\lambda_1\neq0\), \(\lambda_n\nearrow+\infty\)) a sequence of complex numbers satisfying NEWLINE\[NEWLINEn_\Lambda(r)\leqslant r^{\rho(r)}+O(1)\,.\tag{1}NEWLINE\]NEWLINENEWLINENEWLINELet NEWLINE\[NEWLINE\begin{gathered} E_0(w)=1-w\quad\text{and}\quad E_p(w)=(1-w)\exp\left(\sum\limits_{k=1}^p\frac{w^k}k\right),> p\in\mathbb{N}, \\ \mathcal{M}_p(r)=\max\limits_{0\leqslant\varphi\leqslant2\pi}\log|E_p(re^{i\varphi})|,\quad S(\rho)=\int\limits_0^{+\infty}r^{-\rho}\,d\,\mathcal{M}_p(r), \\ f_\Lambda(z)=\prod\limits_{n=1}^\infty E_p\left(\frac z{\lambda_n}\right)>(p=[\rho])\,. \end{gathered}NEWLINE\]NEWLINENEWLINENEWLINEThe author obtains an asymptotically sharp estimate NEWLINE\[NEWLINE \log M(f_\Lambda,R)\leqslant R^{\rho(R)}(S(\rho)+\varepsilon(R)), \quad\varepsilon(R)\to0> (R\to+\infty)\,,\tag{2} NEWLINE\]NEWLINE in the case when the proximate order \(\rho(r)\) in (1) displays a `regular behaviour'. On the other hand, if the proximate order in (1) is arbitrary, then finding the function \(\varepsilon\) providing an asymptotically sharp estimate (2) seems to be a difficult problem in general. So the author defines classes of proximate orders whose union is the whole set of proximate orders and proves the implication \((1)\Rightarrow(2)\) for a function \(\varepsilon\) which is common for the particular class of proximate orders. An upper estimate for the logarithm of the maximum modulus of a canonical product is also proved in the paper. In the case when (1) involves a certain special proximate order from the class under consideration and the arguments of elements of \(\Lambda\) have a certain special distribution it is proved in the paper that this estimate is asymptotically sharp. In the paper the author indicates explicitly the infinitesimal function \(\varepsilon\) in Gold'berg's lower bound NEWLINE\[NEWLINE \log M(F,R_k)\geqslant R_k^{\rho(R_k)}(S(\rho)-\varepsilon(R_k))\tag{3} NEWLINE\]NEWLINE which holds in the case when the counting function of the sequence \(\Lambda\) satisfies the condition NEWLINE\[NEWLINE r^{\rho(r)}->\text{const}\leqslant n(r)=O(r^\tau),\quad\tau\in(p,p+1),>p=[\rho]\,,\tag{4} NEWLINE\]NEWLINE the arguments of elements of \(\Lambda\) have a certain special distribution and \(\{R_k\}\) is a sequence tending to \(+\infty\). Here the proximate order can be arbitrary. Then we observe that inequality (4) cannot be improved significantly in the corresponding classes of proximate orders. Note that estimates for the counting function (1) and (4) are compatible. Thus the estimates (2) and (3) demonstrate that if the counting function of the sequence \(\Lambda\) satisfies (1) and (4) and the arguments of the points in the sequence \(\Lambda\) are distributed as prescribed (in the paper), then the canonical product \(f_\Lambda\) has type \(S(\rho)\) for the proximate order \(\rho(r)\). In the paper the behaviour of \(S(\rho)\) is also studied.